2014-12-94

InterviewYoung people need to show something in order to getahead and get a job.One could make it harder to get papers publishedPublishers would not like that. They thrive on greaterand greater volumes. I do not see how it can be changed.And not that it is such a big problem either, if I am to behonest.You think that quantity by itself can be a good thing?In my field of analysis, you have results which are trueunder certain conditions. What those conditions are cannotbe fixed in any clean canonical way, as is maybe thecase in other fields of mathematics. If you change theconditions, the results become subtly different. There iscertainly great value in exploring this systematically andto do so, you really need a lot of people working. Most ofthe results obtained may not be interesting at all but younever know what you may come up with.This ties up with mathematics becoming more and moreof a big science, something the funding agencies woulddefinitely appreciate.Of course they do. Big projects are what they are comfortablewith. It makes it much easier to give money forone thing. Another tendency which I very much deploreis to identify winners and shower them with money, whileothers do not get anything at all – a clear case of winnerstake it all. I can see that this system may have somerelevance to big science when a project needs a lot ofresources and you have to prioritise in order to maintaincritical masses, but in mathematics? It is so different. Amathematician does not need much money, only enoughto keep him comfortable and not having to worry aboutbasic needs, including such things as going to meetingsthat he may find interesting or inviting people whom hemay want to talk to and learn from. We are clearly talkingabout peanuts.Could it be possible that the forms of mathematical researchwill change, partly under the pressures of funding,and that it will be more like in big science wherethere are large projects involving a lot of people witha definite hierarchy and where most people are simplytold what to do? You are good at a certain type of combinatorialarguments – solve this! Maybe many mathematiciansmight find this a relief, not having to takepersonal responsibility for their research.As to a big project, the only thing I know of is the classificationof finite simple groups, although Polymath providesa systematic attempt to pool the efforts of many toa common goal, where everyone puts in their piece ofthe puzzle but without the hierarchy you are referringto.As one of my colleagues put it, in other fields graduatestudents are an asset, in mathematics a liability.Yes, you have to come up with a good problem for themand more often than not solve it. As to the concept oflarge projects, it is good for some things but when itcomes to the creative breakthroughs in mathematics, thekinds of things we referred to as coming out of the blue,this is solely the result of individual efforts.Another danger with this change of the traditional cultureis that mathematics may be diverted into avenuesthat are not intrinsically interesting from a mathematicalpoint of view. As an example, take the calculationsof the shapes of complex molecules in life sciences.Those shapes can, in principle, be derived from basicquantum physics but it seems that this will have to domore with simulations than mathematical stimulation-- no global understanding. Is there a danger that mathematicswill run out of simple but powerful ideas andbecome inhuman, in the sense of being inaccessible tothe individual mind.It is true we mathematicians prefer to understand whysomething is true, not just being told using some complicatedverification. But this does not only occur in appliedmathematics but also in pure; I think of the notoriouscomputer proof of the four-colour problem fromthe 1970s. The theorem is true just because a computerhas checked a vast number of special cases. It gives noinsight. As to your worry about running out of new excitingthings, this is far from imminent. I am thinking inparticular of the recent results on sparse matrices whichwere presented here in Seoul at the congress.Have you ever read a mathematics book from cover tocover?Come to think of it, I think only once or twice, when Iwas a young student. A good writer of a mathematicalbook knows this of course and writes in such a way that itcan be disassembled into small, self-contained partsWe discussed before the impossibility of giving definiteformulations to theorems. What is important in a resultis not any of its various formulations but the idea thatlies behind it. You cannot treat a theorem as a blackbox. But ideas can never be formulated.I am not so sure about that. I think one can convey ideas,but it is true it has to be done obliquely. You can presentthem many times, subtly changing the formulations, sayingthe same thing over and over……This ties in with the often touted opinion that mathematicsis something you get used to. You may not reallyunderstand what you are doing but believe you do……Another very important thing is to present the instructiveexample. The specific example conveys the generalidea without having to formulate it. In my talk I gave anexplicit example; did you go to my talk?No, I was unable to do so. In the same vein it seems thatthe most effective ways of conveying mathematical understandingis through personal conversation – the timehonoured method since the beginning of human time.Why is that really?It has to do with the pacing, the possibility of direct interaction.You can ask questions. Pictures can be drawn.46 EMS Newsletter December 2014